This study will determine whether MK-4827 can be safely administered in combination with pegylated liposomal doxorubicin, and if so, will obtain an estimate of the benefit of the combination in patients with ovarian cancer as compared to historical data with single agent pegylated liposomal doxorubicin. The first part of the study (Part A) is designed to determine the maximum tolerated dose (MTD) and evaluate the safety of MK-4827, when administered in combination with pegylated liposomal doxorubicin. Part B is designed to assess preliminary clinical activity of MK-4827, when administered in combination with pegylated liposomal doxorubicin to participants with ovarian cancer. It is hypothesized that MK-4827 can be administered, in conjunction with pegylated liposomal doxorubicin, with acceptable tolerability and that MK-4827, administered in conjunction with pegylated liposomal doxorubicin, will demonstrate a tumor response rate equal or superior to that of historical data for pegylated liposomal doxorubicin alone.